Higher June 2019 Paper 1R Q24

EdexcelHigherCurrent spec5 marksVectors

Quadrilateral OABC with AB parallel to OC; vector a along OA and vector c along OC; P marked on AB and Q marked on AC

Diagram NOT accurately drawn

\(\overrightarrow{OA} = \mathbf{a}\) \(\overrightarrow{OC} = \mathbf{c}\) \(\overrightarrow{AB} = 2\mathbf{c}\)

\(P\) is the point on \(AB\) such that \(AP : PB = 3 : 1\)
\(Q\) is the point on \(AC\) such that \(OQP\) is a straight line.
Use a vector method to find \(AQ : QC\)
Show your working clearly.

(5)

Mark scheme

Scheme	Marks
\(\overrightarrow{AP} = \dfrac{3}{4} \times 2\mathbf{c}\) (\(= \dfrac{3}{2}\mathbf{c}\)) oe	M1
\(\overrightarrow{AC} = \mathbf{c} - \mathbf{a}\) oe or \(\overrightarrow{CA} = \mathbf{a} - \mathbf{c}\) oe	M1
\(\overrightarrow{OQ} = \mathbf{c} + n(\mathbf{a} - \mathbf{c})\) or \(\overrightarrow{OQ} = \mathbf{a} + n(\mathbf{c} - \mathbf{a})\) or \(\overrightarrow{QP} = n(\mathbf{a} - \mathbf{c}) + \dfrac{3}{2}\mathbf{c}\)	M1
\(\dfrac{n}{1 - n} = \dfrac{2}{3} \Rightarrow n = \dfrac{2}{5}\) oe or \(\dfrac{1 - n}{n} = \dfrac{2}{3} \Rightarrow n = \dfrac{3}{5}\) oe or \(\dfrac{n}{\frac{3}{2} - n} = \dfrac{2}{3} \Rightarrow n = \dfrac{3}{5}\) oe	M1
Working required Answer: 3 : 2	A1
	(5)
	(5 marks)

Notes

M1: For \(\overrightarrow{AP} = \dfrac{3}{2}\mathbf{c}\) oe, eg could be part of \(\overrightarrow{OP} = \mathbf{a} + \dfrac{3}{2}\mathbf{c}\) oe or on diagram

A1: oe, dep on M3